Appropriate use of the term "transversely"

In the following paragraph, is it appropriate to use the term "transversely" to describe something that has the opposite effect?

Tests have shown that the lower the range, the more likely that a submatrix will be singular. Transversely, the larger the range, the least likely that a submatrix will be singular.

If not, is there another word that I can use instead? I know that, in this instance, I cannot use the word "conversely" as this means something totally different.

No, transversely is incorrect. You might consider "In contrast ..." or "On the other hand...".

I would use "inversely", as in "inversely proportional".

If we have a statement:

The lower the range, the more likely that a submatrix will be singular.

the logical converse of the statement is:

The more likely that a submatrix will be singular, the lower the range.

and the logical inverse of the statement is:

The larger the range, the less likely that a submatrix will be similar.

So I would say that, if your aim is logical correctness, you should use the adverb inversely. This usage is also compatible with the common English meaning of the word inversely. The adverb transversely is just wrong here. From Merriam-Webster online dictionary:

Transversely: in a line or direction running from corner to corner.

But never use the adverb contrapositively. It doesn't exist.